A wheel of radius `1 m` rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially on contact with the ground is.

Options

A. `pi`
B. `2 pi`
C. `sqrt(2) pi`
D. `sqrt(pi^2 + 4)`

Explanation

To solve the problem of finding the magnitude of the displacement of the point of the wheel initially in contact with the ground after it rolls forward half a revolution, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Position**:
The wheel has a radius of `1 m`. Initially, the point of contact with the ground is at the bottom of the wheel.

2. **Determine the Final Position After Half a Revolution**:
After rolling half a revolution, the point that was initially in contact with the ground will move to the top of the wheel. The distance traveled by the wheel along the ground during this half revolution is equal to half the circumference of the wheel.

3. **Calculate the Distance Traveled**:
The circumference of the wheel is given by the formula:
\[ C = 2\pi r \]
For a radius of `1 m`, the circumference is:
\[ C = 2\pi \times 1 = 2\pi \text{ m} \]
Therefore, the distance traveled during half a revolution is:
\[ \text{Distance} = \frac{1}{2} \times C = \frac{1}{2} \times 2\pi = \pi \text{ m} \]

4. **Visualize the Displacement**:
The initial position of the point is at `(0, 0)` (the bottom of the wheel), and after half a revolution, it moves to the top of the wheel at `(0, 2)` (since the height of the wheel is equal to its diameter, which is `2 m`).

5. **Calculate the Displacement**:
The displacement is the straight-line distance from the initial position to the final position. Using the coordinates:
- Initial position: `(0, 0)`
- Final position: `(π, 2)`

The displacement can be calculated using the Pythagorean theorem:
\[ \text{Displacement} = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]
where:
- \(\Delta x = \pi - 0 = \pi\)
- \(\Delta y = 2 - 0 = 2\)

Therefore, the displacement is:
\[ \text{Displacement} = \sqrt{(\pi)^2 + (2)^2} = \sqrt{\pi^2 + 4} \]

6. **Final Answer**:
The magnitude of the displacement of the point of the wheel initially in contact with the ground after rolling forward half a revolution is:
\[ \sqrt{\pi^2 + 4} \text{ m} \]